Quantum Computer, Non-Transitory Computer Readable Media Storing Program, Quantum Calculation Method, And Quantum Circuit

ABSTRACT

A quantum computer includes: a setting unit configured to set a parameter group of n layers based on each coefficient in a linear sum of unitary operators whose number is 2 to the n-th power, wherein the parameter group of k-th (2≤k≤n) layer is recursively set based on the parameter group of (k−1)-th layer; a quantum gate having n+m qubits including n auxiliary qubits and m target qubits, and configured to execute a predetermined calculation on an input value input to each qubit based the parameter group of n layers; and a specification unit configured to specify the linear sum of the unitary operators based on a calculation result of the quantum gate.

CROSS REFERENCE TO RELATED APPLICATIONS

The present application claims priority under 35 U.S.C. § 119 toJapanese Patent Application No. 2020-110658, filed Jun. 26, 2020.

BACKGROUND Technical Field

The present invention relates to a quantum computer, a non-transitorycomputer readable media storing a program, a quantum calculation method,and a quantum circuit.

Related Art

Computers that used quantum mechanics for information processing wereproposed in the 1980s, and various quantum information processingtechnologies are still being proposed presently.

For instance, Japan unexamined Patent Application Publication No.2015-135377 discloses a prior art of a quantum computer and a quantumcalculation method.

However, some physical quantities and physical functions include linearsums of unitary operators. Although unitary operators are generallyhandled in quantum calculation, since there is no method for quantumcalculation of a generic linear sum of unitary operators, it isnecessary to calculate such physical quantities and physical functions,and a classical algorithm must be used. For this reason, a classicalcomputer is used, but since the classical computer has a limit in theamount of calculation and the calculation speed, this can be a barrierto calculation.

SUMMARY OF INVENTION

In view of the above circumstances, the present invention provides atechnique for quantum computation of the generic linear sum of unitaryoperators.

The present invention provides a quantum computer, comprising: a settingunit configured to set a parameter group of n layers based on eachcoefficient in a linear sum of unitary operators whose number is 2 tothe n-th power, wherein the parameter group of k-th (2≤k≤n) layer isrecursively set based on the parameter group of (k−1)-th layer; aquantum gate having n+m qubits including n auxiliary qubits and m targetqubits, and configured to execute a predetermined calculation on aninput value input to each qubit based on the parameter group of nlayers; and a specification unit configured to specify the linear sum ofthe unitary operators based on a calculation result of the quantum gate.

Since the linear sum of the unitary operators can be calculated withquantization, physical quantities and physical functions including thelinear sum of the unitary operators can be calculated instead of theclassical algorithm in such a quantum computer.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram showing a hardware configuration of a quantumcomputer 1.

FIG. 2 is a block diagram showing a functional configuration of aquantum computer 1 (quantum processor 5).

FIG. 3 shows the configuration of the quantum gate 53 (quantum circuit).

FIG. 4 is a circuit diagram showing processing of a parameter group {θ}.

FIG. 5 is a circuit diagram showing processing of a parameter group {θ}when n=2.

FIG. 6 is a circuit diagram for deriving the electronic creationoperator a + and the annihilation operator a of the Green's function GF.

FIG. 7 is a graph showing the calculation result of the quasiparticlespectrum of the H_2O molecule.

FIG. 8 is a circuit diagram for deriving the electronic density operatorn of the linear response function LF.

FIG. 9 is a circuit diagram for deriving the spin density operator s ofthe linear response function LF.

FIG. 10 is a graph showing a calculation result of a light absorptionspectrum of a C_2 molecule.

FIG. 11 shows an example of various quantum circuits related to oracle.

DETAILED DESCRIPTION OF EMBODIMENTS

Hereinafter, an embodiment of the present invention will be describedwith reference to the drawings. Various features described in theembodiment below can be combined with each other.

Further, in the present embodiment, the “unit” may include, for example,a combination of hardware resources implemented by a circuit in a broadsense and information processing of software that can be concretelyrealized by these hardware resources. In addition, various informationis handled in this embodiment, and these information are represented byquantum superposition (so-called qubits), and communication/operationcan be executed on a circuit in a broad sense including a memory 4 and aquantum processor 5.

1. Hardware Configuration

In this section, the hardware configuration of a quantum computer 1according to the present embodiment will be described.

FIG. 1 is a block diagram showing the hardware configuration of thequantum computer 1. As shown in FIG. 1, the quantum computer 1 includesa communication unit 3, a memory 4, and a quantum processor 5. Thesecomponents are connected to each other inside the quantum computer 1 viaa communication bus 2. Hereinafter, each component will be furtherdescribed.

(Communication Unit 3)

The communication unit 3 is used for the quantum computer 1 to performinformation communication with another information processing device (aclassical computer, a quantum computer, or a computer combines them) ora peripheral device.

(Memory 4)

The memory 4 stores various information defined by the abovedescription. In particular, the memory 4 stores various programs thatcan be read by the quantum processor 5 described below. Further, thememory 4 stores physical property information and the like of a specificmaterial related to the calculation of the quantum computer 1 asnecessary.

(Quantum Processor 5)

The quantum processor 5 processes and controls the overall operationrelated to the quantum computer 1. The quantum processor 5 realizesvarious functions related to the quantum computer 1 by reading a programstored in the memory 4 or a predetermined program input via thecommunication unit 3. That is, information processing by software(stored in the memory 4 or input from another information processingdevice via the communication unit 3) is specifically realized by thehardware (quantum processor 5). Then, it can be executed as eachfunctional unit as shown in FIG. 2. These will be described in detail inthe next section. Although it is represented as a single quantumprocessor 5 in FIG. 1, there is no limitation to this, and it may beimplemented so as to have a plurality of quantum processors 5 for eachfunction. Moreover, it may be a combination thereof.

2. Functional Configuration

In this section, the functional configuration of the quantum computer 1according to the present embodiment will be described. FIG. 2 is a blockdiagram showing a functional configuration of a quantum computer 1(quantum processor 5). FIG. 3 shows the configuration of the quantumgate 53 (quantum circuit). Regarding the quantum processor 5 describedabove, the quantum computer 1 includes a setting unit 51, a processingunit 52, a quantum gate 53, an observation unit 54, a quantum phaseestimation unit 55, and a specification unit 56. Hereinafter, eachcomponent will be further described.

(Setting Unit 51)

When the quantum computer 1 calculates the linear sum O of the unitaryoperators U, the linear sum O is defined by the number of terms being 2to the n-th power. A term having a coefficient c of 0 is also includedin this number of terms. The setting unit 51 is configured to set theparameter group {θ} of the n layers based on each coefficient c in thelinear sum O of the unitary operators U. Here, the parameter group {θ}of the k-th (2≤k≤n) layer is recursively set based on the parametergroup {θ} of the (k−1)-th layer.

(Processing Unit 52)

The processing unit 52 executes various calculations calculated by thequantum computer 1. For instance, the processing unit 52 executes thefour arithmetic operations included in the quantum calculation.Alternatively, the processing unit 52 may be configured so that theinput value to be input to the quantum gate 53, which will be describedlater, can be initialized. For instance, the processing unit 52 caninitialize the input value of the auxiliary qubit AQ by causing theHadamard gate (H in FIG. 3) to act on all the qubits as an initialstate. The input value of the initialized auxiliary qubit AQ has aplurality of levels with equal probability.

(Quantum Gate 53)

As shown in FIG. 3, the quantum gate 53 has n+m qubits including nauxiliary qubits AQ and m target qubits TQ. The quantum gate 53 isconfigured to perform a predetermined operation on the input value inputto each qubit based on the parameter θ included in the parameter group{θ} of the n layers.

More specifically, the quantum gate 53 (quantum circuit) includes aninput unit 531, a calculation unit 532, and an output unit 533(particularly, a target qubit output unit 533 a). The input unit 531 hasn+m qubits including n auxiliary qubits AQ and m target qubits TQ. Thecalculation unit 532 is configured to execute a predeterminedcalculation on the input value input from the input unit 531 based onthe parameter θ included in the parameter group {θ} of the n layers. Thetarget qubit output unit 533 a is configured to output the product ofthe input value input to the target qubit TQ and the linear sum O of theunitary operators U.

(Observation Unit 54)

The observation unit 54 observes an output value having a plurality ofvalues probabilistically by quantum superposition for each qubit. Byobserving by the observation unit 54, the output value is determined tobe one of a plurality of values. Specifically, the observation unit 54is configured to observe the auxiliary output value which is the outputvalue of the n auxiliary qubits AQ in the quantum gate 53. In thisregard, as shown in FIG. 3, the quantum gate 53 is configured to outputthe product of the input value of the target qubit TQ and the linear sumO of the unitary operators U as the target output value which is theoutput value of the target qubit TQ when the auxiliary output valueobserved by the observation unit 54 is a certain value (for example,when all are 0).

More preferably, the quantum gate 53 outputs the product of the inputvalue of the target qubit TQ and the first linear sum O1 of the unitaryoperators U as the first target output value when the observation unit54 observes the auxiliary output value as 0. On the other hand, thequantum gate 53 outputs the product of the input value of the targetqubit TQ and the second linear sum O2 of the unitary operators U as thesecond target output value when the observation unit 54 observes theauxiliary output value as 1. Here, the second linear sum O2 is theHermitian conjugate of the first linear sum O1. This will be describedin detail in the next section.

More preferably, the quantum gate 53 is configured to output the stateof any quantum qubit by outputting the product of the input value of thetarget qubit TQ and the linear sum O of the unitary operators U as astandard output value when the observation unit 54 observes theauxiliary output value as 0. This will be described in detail in thenext section.

(Quantum Phase Estimation Unit 55)

The quantum phase estimation unit 55 is configured to perform quantumphase estimation for the first and second target output values. Thedetails will be illustrated in the next section as well.

(Specification Unit 56)

The specification unit 56 is configured to specify the linear sum O ofthe unitary operators U based on the calculation result of the quantumgate 53. Further, the specification unit 56 may be configured to specifya value or a function that depends on the linear sum O of the unitaryoperators U. For example, the specification unit 56 is configured tospecify a predetermined function based on the result of the quantumphase estimation by the quantum phase estimation unit 55. Here, examplesof the predetermined function include Green's function GF or linearresponse function LF.

3. Theory

In this section, the quantum mechanical theory regarding theabove-mentioned functional configuration will be supplemented, and thequantum calculation method based on this will be illustrated.

3.1 Recursive Setting of Parameter Group {θ}

As described in the previous section, the setting unit 51 recursivelysets the parameter group {θ} so as to calculate the linear sum O of theunitary operators U. As a premise, regarding the linear sum O of theunitary operators U whose number of terms is defined by 2 to the n-thpower, each coefficient c is sequentially identified using an indexrepresented by an n-digit binary number. This is represented as inEquation 1.

[Equation 1]

=c _(0 . . . 00) U _(0 . . . 00) +c _(0 . . . 01) U _(0 . . . 01) + . .. +c _(1 . . . 11) U _(1 . . . 11)  Eqn. 1

Subsequently, the parameter θ constituting the parameter group {θ} isdefined based on each coefficient c. The phase factor of c_λ (λ is asubscript and represents a sequence of numbers 0 and 1) is set as θ_λ.(That is, c_λ=|c_λ| exp(iθ_λ).) Hereinafter, c_λ is retaken as |c_λ| andc_λ is a positive real number. Further, the parameter group {θ} has nlayers, and the parameter group {θ} of the k-th layer includes 2parameters θ{circumflex over ( )}(k) raised to the n−k-th power. Here,as shown in Equation 2, the tangent of each parameter θ{circumflex over( )}(k) is expressed so as to depend on the ratio of a set of specificcoefficients c. That is, the setting unit 51 determines the parameterθ{circumflex over ( )}(k) included in the parameter group {θ} of thek-th layer so that the parameter θ{circumflex over ( )}(k) depends onthe ratio of a specific set of coefficients c out of each coefficient c.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 2} \right\rbrack & \; \\\left\{ \begin{matrix}{{\tan\theta}_{0\;.\;.\;.\; 00}^{(1)} = \frac{c_{0\;.\;.\;.\; 001}}{c_{0\;.\;.\;.\; 000}}} \\{{\tan\theta}_{0\;.\;.\;.\; 01}^{(1)} = \frac{c_{0\;.\;.\;.\; 011}}{c_{0\;.\;.\;.\; 010}}} \\\vdots \\{{\tan\theta}_{0\;.\;.\;.\; 11}^{(1)} = \frac{c_{1\;.\;.\;.\; 111}}{c_{1\;.\;.\;.\; 110}}}\end{matrix} \right. & {{Eqn}.\mspace{14mu} 2}\end{matrix}$

Here, the specific set of coefficients c are two coefficients c havingindices which differ from each other only in the k-th lowest digit,while the other digits coincide with each other. For instance, for theparameter θ included in the parameter group {θ} of the first layer,θ{circumflex over ( )}(1)_{0 . . . 00} depends on the value obtained bydividing c_{0 . . . 001} by c_{0 . . . 000}. However, these coefficientsc have the indices which differ from each other only in the lowestdigit. Further, regarding the parameter θ included in the parametergroup {θ} of the second layer, θ{circumflex over ( )}(2)_{0 . . . 00}depends on the value obtained by dividing c_{0 . . . 010} by c_{0 . . .000}. However, these coefficients c have the indices which differ fromeach other only in the second lowest digit (see Equation 3).

FIG. 4 is a circuit diagram showing the processing of the parametergroup {θ}. j in FIG. 4 may be considered as j=k+1. H is a Hadamard gatethat performs the Hadamard transform. R is a rotation operator of theBloch sphere and is represented by Ry(−2θ)=exp(iθσ_y). U is a unitaryoperator that performs unitary transformation. The black circle causesthe value of the corresponding target qubit (generally meant here) toperform a predetermined operation when the input value of the controlqubit is 1, and does not manipulate the value of the correspondingtarget qubit when the input value of the control qubits 0. The whitecircle causes the value of the corresponding target qubit to perform apredetermined operation when the input value of the control qubits 0,and does not manipulate the value of the corresponding target qubit whenthe input value of the control qubits 1. The coefficient c related tothe parameter θ{circumflex over ( )}(1) of the first layer isrepresented by the equivalent circuit 532 a when regarded as a gateoperation. The coefficient c related to the parameter θ{circumflex over( )}(k) of the second and subsequent layers is represented by using thecoefficient c related to the parameter θ{circumflex over ( )}(k−1) ofthe next lower layer when regarded as a gate operation. Specifically, itis defined as in Equation 3.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 3} \right\rbrack & \; \\\left\{ \begin{matrix}{{\frac{{\cos\theta}_{0\;.\;.\;.\; 001}^{(1)}}{{\cos\theta}_{0\;.\;.\;.\; 000}^{(1)}}{\tan\theta}_{0\;.\;.\;.\; 00}^{(2)}} = \frac{c_{0\;.\;.\;.\; 010}}{c_{0\;.\;.\;.\; 0000}}} \\{{\frac{{\cos\theta}_{0\;.\;.\;.\; 011}^{(1)}}{{\cos\theta}_{0\;.\;.\;.\; 010}^{(1)}}{\tan\theta}_{0\;.\;.\;.\; 01}^{(2)}} = \frac{c_{0\;.\;.\;.\; 0110}}{c_{0\;.\;.\;.\; 0100}}} \\\vdots \\{{\frac{{\cos\theta}_{1\;.\;.\;.\; 111}^{(1)}}{{\cos\theta}_{1\;.\;.\;.\; 110}^{(1)}}{\tan\theta}_{1\;.\;.\;.\; 11}^{(2)}} = \frac{c_{1\;.\;.\;.\; 1110}}{c_{1\;.\;.\;.\; 1100}}}\end{matrix} \right. & {{Eqn}.\mspace{14mu} 3}\end{matrix}$

Regarding Equation 3, assuming that the parameter θ included in theparameter group {θ} of the k-th layer is identified in order by using anindex represented by an (n−k)-digit binary number, the setting unit 51sets the parameter θ so that a first parameter θ{circumflex over ( )}(k)included in the parameter group of the k-th layer depends on the ratioof cosines for a particular set of the second parameters θ{circumflexover ( )}(k−1) included in the parameter group {θ} of the (k−1)-thlayer. Here, the particular set of the second parameters θ{circumflexover ( )}(k−1) consists of two parameters θ, each of which has an indexwith 0 or 1 appended to the end of that of the first parameterθ{circumflex over ( )}(k).

The setting of such a parameter group {θ} is represented as in Equation4.

$\begin{matrix}\left. {{\left. {{\left. {{\left. {{{\overset{\overset{Ancillae}{︷}}{\left. {{{\left. {q_{n - 1}^{A} = 0} \right\rangle \otimes}}0} \right\rangle^{\otimes {({n - 1})}}} \otimes \overset{\overset{Input}{︷}}{\left. \psi \right\rangle}}\overset{H_{n}}{\mapsto}{{\frac{\left. {{{\left. 0 \right\rangle +}}1} \right\rangle}{\sqrt{2}} \otimes H_{n - 1}}\left. \Phi_{n - 1} \right\rangle}\overset{{cc}_{0}^{({n - 1})}\mspace{14mu}{and}\mspace{14mu}{cc}_{1}^{({n - 1})}}{\mapsto}{{\frac{\left. {❘0} \right\rangle}{\sqrt{2}} \otimes C_{0}^{({n - 1})}}H_{n - 1}}}❘\Phi_{n - 1}} \right\rangle + {{\frac{\left. {❘1} \right\rangle}{\sqrt{2}} \otimes C_{1}^{({n - 1})}}H_{n - 1}}}❘\Phi_{n - 1}} \right\rangle\overset{R_{y}}{\mapsto}{{\frac{\left. {0,\theta^{(n)}} \right\rangle}{\sqrt{2}} \otimes C_{0}^{({n - 1})}}H_{n - 1}}}❘\Phi_{n - 1}} \right\rangle + {{\frac{\left. {1,\theta^{(n)}} \right\rangle}{\sqrt{2}} \otimes C_{0}^{({n - 1})}}H_{n - 1}}}❘\Phi_{n - 1}} \right\rangle & {{Eqn}.\mspace{14mu} 4}\end{matrix}$

Further, when this is recursively expanded, it is expressed as inEquation 5 when all the auxiliary output values of the auxiliary qubitsAQ are observed to be 0.

$\begin{matrix}{\mspace{79mu}\left\lbrack {{Equation}\mspace{14mu} 5} \right\rbrack} & \; \\{{\left. {\left. {C^{(n)}H_{n}} \middle| \Phi_{n} \right\rangle = \left| 0 \right.} \right\rangle^{\otimes n} \otimes {\frac{1}{2^{n/2}}\left\lbrack {{{\cos\theta}^{(n)}{\cos\theta}_{0}^{({n - 1})}{\cos\theta}_{00}^{({n - 2})}{\cdots cos\theta}_{0{\cdots 0}}^{(2)}{\cos\theta}_{0{\cdots 00}}^{(1)}U_{0{\cdots 000}}} + {{\cos\theta}^{(n)}{\cos\theta}_{0}^{({n - 1})}{\cos\theta}_{00}^{({n - 2})}{\cdots cos\theta}_{0{\cdots 0}}^{(2)}{\sin\theta}_{0{\cdots 00}}^{(1)}U_{0{\cdots 001}}} + \mspace{200mu}{\vdots{\sin\theta}^{(n)}{\sin\theta}_{1}^{({n - 1})}{\sin\theta}_{11}^{({n - 2})}{\cdots sin\theta}_{1{\cdots 1}}^{(2)}{\cos\theta}_{1{\cdots 11}}^{(1)}U_{1{\cdots 110}}} + \left. \quad\left. \left. \quad{{\sin\theta}^{(n)}{\sin\theta}_{1}^{({n - 1})}{\sin\theta}_{11}^{({n - 2})}{\cdots sin\theta}_{1{\cdots 1}}^{(2)}{\sin\theta}_{1{\cdots 11}}^{(1)}U_{1{\cdots 111}}} \right\rbrack \middle| \psi \right. \right\rangle}\quad \right.}} + {{\quad\quad}\left( {{terms}\mspace{14mu}{involving}\mspace{14mu}{other}\mspace{14mu}{ancillary}\mspace{14mu}{states}} \right)}} & {{Eqn}.\mspace{14mu} 5}\end{matrix}$

That is, the unitary transformation of the k-th layer is recursivelyrepresented by the unitary transformation of the (k−1)-th layer. Morespecifically, the unitary transformation of the k-th layer is determinedas follows. Referring to FIG. 4, the auxiliary qubit AQ0 to theauxiliary qubit AQ(k−1) are set to the input values 0, respectively, andthe initialization state in which 0 and 1 are superpositioned as thebasis is realized through the Hadamard gate. Next, when the auxiliaryqubit AQ(k−1) is 0, the first unitary transformation of the (k−1)-thlayer for the auxiliary qubit AQ0 to the auxiliary qubit AQ(k−2) and mtarget qubits is executed. Subsequently, when the auxiliary qubitAQ(k−1) is 1, the second unitary transformation of the (k−1)-th layerfor the auxiliary qubit AQ0 to the auxiliary qubit AQ(k−2) and m targetqubits is executed. Here, in the first unitary transformation and thesecond unitary transformation, their indices differ from each other onlyin the lowermost digit. Finally, the rotation by the rotation operatorRy of θ{circumflex over ( )}(k) is executed for the auxiliary qubitAQ(k−1).

In other words, the quantum gate 53 includes n layers control unitarygate. The control unitary gate of the k-th (2≤k≤n) layer inputs onecontrol qubit, k−1 controlled qubits, and m target qubits. A first andsecond control unitary gates of the (k−1)-th layer and a rotation gateare provided. The first control unitary gate is configured to operatethe values of the controlled qubit and the target qubit when the valueof the control qubits 0. The second control unitary gate is configuredto operate the values of the controlled qubit and the target qubit whenthe value of the control qubits 1. The rotation gate applies on thecontrol qubit.

3.2 Interpretation of a Simple Model

Next, consider the case of n=2 with reference to FIG. 5. FIG. 5 is acircuit diagram showing processing of the parameter group {θ} when n=2.The linear sum O of the four unitary operators U included in FIG. 5 isshown in Equation 6.

[Equation 6]

=c ₀₀ U ₀₀ +c ₀₁ U ₀₁ +c ₁₀ U ₁₀ +c ₁₁ U ₁₁  Eqn. 6

As shown in FIG. 5, the quantum gate 53 a has two auxiliary qubits AQand m target qubits TQ. In this quantum circuit, the parameter group {θ}includes the parameter group {θ} of the first layer and the parametergroup {θ} of the second layer. The parameter group {θ} of the firstlayer includes θ{circumflex over ( )}(1)_{0} and θ{circumflex over( )}(1)_{1}. Then, the tangent of θ{circumflex over ( )}(1)_{0} isdefined by the value of c_{01} divided by c_{00}. The tangent ofθ{circumflex over ( )}(1)_{1} is defined by the value of c_{11} dividedby c_{10}. The parameter group {θ} of the second layer includesθ{circumflex over ( )}(2) and has no index. Then, the product of thetangent of θ{circumflex over ( )}(2) and the ratio of cosines string forθ{circumflex over ( )}(1)_{0} and θ{circumflex over ( )}(1)_{1} isdefined by the value of c_{10} divided by c_{00}. That is, as shown inEquation 7, θ{circumflex over ( )}(2) included in the parameter group{θ} of the second layer is recursively set by θ{circumflex over( )}(1)_{0} and θ{circumflex over ( )}(1)_{1} included in the parametergroup {θ} of the first layer.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 7} \right\rbrack & \; \\{\;\left\{ \begin{matrix}{{\tan\theta}_{0}^{(1)} = \frac{c_{01}}{c_{00}}} \\{{\tan\theta}_{1}^{(1)} = \frac{c_{11}}{c_{10}}} \\\; \\{{\frac{{\cos\theta}_{1}^{(1)}}{{\cos\theta}_{0}^{(1)}}{\tan\theta}^{(2)}} = \frac{c_{10}}{c_{00}}}\end{matrix} \right.} & {{Eqn}.\mspace{14mu} 7}\end{matrix}$

In such a quantum circuit, when all the auxiliary output values in thetwo auxiliary qubits AQ are observed to be 0, the target output value inthe target qubits TQ is the product of the input value of the targetqubit TQ and the linear sum O of the unitary operators U shown inEquation 6.

More specifically, referring to FIG. 5, the auxiliary qubit AQ0 and theauxiliary qubit AQ1 are set to the input values 0, respectively, and theinitialization state in which 0 and 1 are superpositioned as the basisis realized through the Hadamard gate. Next, when the auxiliary qubitAQ1 is 0 and the auxiliary qubit AQ0 is 0, unitary transformation by theunitary operators U_{00} is executed for the target qubit TQ. Then, whenthe auxiliary qubit AQ1 is 0 and the auxiliary qubit AQ0 is 1, unitarytransformation by the unitary operators U_{01} is executed for thetarget qubit TQ. Subsequently, when the auxiliary qubit AQ1 is 0, therotation by the rotation operator Ry of θ{circumflex over ( )}(1)_0 isexecuted for the auxiliary qubit AQ0.

Subsequently, unitary transformation by the unitary operators U_{10} isexecuted for the target qubits TQ when the auxiliary qubit AQ1 is 1 andthe auxiliary qubit AQ0 is 0. Then, unitary transformation by theunitary operators U_{11} is executed for the target qubit TQ when theauxiliary qubit AQ1 is 1 and the auxiliary qubit AQ0 is 1. Next, therotation by the rotation operator Ry of θ{circumflex over ( )}(1)_1 isexecuted for the auxiliary qubit AQ0 when the auxiliary qubit AQ1 is 1.

Finally, the rotation by the rotation operator Ry of θ{circumflex over( )}(2) is executed for the auxiliary qubit AQ1.

To summarize the above, this quantum calculation method includes asetting step, a calculation step, and a specification step. In thesetting step, the parameter group {θ} of the n layers is set based oneach coefficient in the linear sum O of the unitary operators whosenumber is 2 to the n-th power. In the calculation step, a predeterminedcalculation is executed for the input value input to each qubit based onthe parameter θ included in the parameter group {θ} of the n layers. Inthe specific step, the linear sum O of the unitary operators U isspecified based on the calculation result in the calculation step.

Preferably, this quantum calculation method further comprises anobservation step. The observation step observes the auxiliary outputvalue which is the output value of n auxiliary qubits AQ in the quantumgate 53. The calculation step outputs the product of the input value ofthe target qubit TQ and the linear sum O of the unitary operators U asthe target output value which is the output value of the target qubit TQwhen all the auxiliary output values observed in the observation stepare 0.

3.3 Application to the Derivation of Green's Function GF

By making it possible to derive the linear sum O of the unitaryoperators U described above, values and functions that depend on thelinear sum O of the unitary operators U can be specified. Here, theGreen's function GF will be described as an example. Once the Green'sfunction GF is obtained, it becomes possible to calculate thequasiparticle band structure (orbital energy). In order to calculate theGreen's function, it is necessary to prepare a state in which theelectronic creation operator a+ (here, “+” is a dagger) and theannihilation operator a act on the basis electronic state. FIG. 6 is acircuit diagram for deriving the diagonal and off-diagonal components ofthe Green's function GF.

The Green's function GF is defined by Equation 8.

[Equation 8]

G(z)=G ^((e))(z)+G ^((h))(z)  Eqn. 8

In Equation 8, the first term is the electronic part and the second termis the hole part. Further, the electronic part of the first term isrepresented by Equation 9, and the hole part of the second term isrepresented by Equation 10.

$\begin{matrix}{\mspace{79mu}\left\lbrack {{Equation}\mspace{14mu} 9} \right\rbrack} & \; \\{{G_{{mm}^{\prime}}^{(e)}(z)} = {\left\langle \Psi_{gs}^{N} \middle| {a_{m}\frac{1}{z + ɛ_{gs}^{N} - \mathcal{H}}a_{m^{\prime}}^{\dagger}} \middle| \Psi_{gs}^{N} \right\rangle = {\sum\limits_{\lambda \in {N + 1}}\frac{\left\langle \Psi_{gs}^{N} \middle| a_{m} \middle| \Psi_{\lambda}^{N + 1} \right\rangle\left\langle \Psi_{\lambda}^{N} \middle| a_{m^{\prime}}^{\dagger} \middle| \Psi_{gs}^{N} \right\rangle}{z + ɛ_{gs}^{N} - ɛ_{\lambda}^{N + 1}}}}} & {{Eqn}.\mspace{14mu} 9} \\{\mspace{79mu}\left\lbrack {{Equation}\mspace{14mu} 10} \right\rbrack} & \; \\{{G_{{mm}^{\prime}}^{(h)}(z)} = {\left\langle \Psi_{gs}^{N} \middle| {a_{m^{\prime}}^{\dagger}\frac{1}{z - ɛ_{gs}^{N} + \mathcal{H}}a_{m}} \middle| \Psi_{gs}^{N} \right\rangle = {\sum\limits_{\lambda \in {N - 1}}\frac{\left\langle \Psi_{gs}^{N} \middle| a_{m^{\prime}}^{\dagger} \middle| \Psi_{\lambda}^{N - 1} \right\rangle\left\langle \Psi_{\lambda}^{N - 1} \middle| a_{m} \middle| \Psi_{gs}^{N} \right\rangle}{z + ɛ_{\lambda}^{N - 1} - ɛ_{gs}^{N}}}}} & {{Eqn}.\mspace{14mu} 10}\end{matrix}$

In particular, a+ in Equation 9 and a in Equation 10 are electroniccreation operator and annihilation operator, respectively. In a quantumcomputer, the above definition formula cannot be calculated as it is.Specifically, even if the wave function in the basis state shown inEquation 11 can be calculated, it is difficult to calculate the state ofthe Equation 12 which is the product of the wave function and thecreation operator a+ or the state of the Equation 13 which is theproduct of the annihilation operator a in a realistic calculation time.

[Equation 11]

|Ψ_(gs) ^(N)

  Eqn. 11

[Equation 12]

a _(m) ^(†)Ψ_(gs) ^(N)

  Eqn. 12

[Equation 13]

a _(m)|Ψ_(gs) ^(N)

  Eqn. 13

Since the electronic creation operator a+ is a Hermitian conjugate ofthe electronic annihilation operator a, the unitary operators U as shownin Equation 14 can be defined by taking the sum or difference thereof.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 14} \right\rbrack & \; \\\left\{ \begin{matrix}{{a^{\dagger} + a} = {\hat{U}}_{0}} \\{{i\left( {a - a^{\dagger}} \right)} = {\hat{U}}_{1}}\end{matrix} \right. & {{Eqn}.\mspace{14mu} 14}\end{matrix}$

When this is solved for the creation operator a+ and the annihilationoperator a, the creation operator a+ and the annihilation operator a arerepresented as the linear sum O of the unitary operator U, respectively.In other words, the first linear sum O1 is the electronic annihilationoperator a in a predetermined function (Green's function GF), and thesecond linear sum O2 is the electronic creation operator a+ in apredetermined function.

$\begin{matrix}\left\lbrack {{Equation}{\mspace{11mu}\;}15} \right\rbrack & \; \\\left\{ \begin{matrix}{a^{\dagger} = {\frac{1}{2}\left( {{\hat{U}}_{0} - {i{\hat{U}}_{1}}} \right)}} \\{a = {\frac{1}{2}\left( {{\hat{U}}_{0} + {i{\hat{U}}_{1}}} \right)}}\end{matrix} \right. & {{Eqn}.\mspace{14mu} 15}\end{matrix}$

That is, by applying the above-mentioned quantum gate 53, the electroniccreation operator a+ and the electronic annihilation operator a can bederived. Specifically, it is preferable to define the quantum gate 53 bfor the diagonal component and the quantum gate 53 c for theoff-diagonal component. In particular, the diagonal components arerepresented as in Equation 16.

$\begin{matrix}{\mspace{79mu}\left\lbrack {{Equation}\mspace{14mu} 16} \right\rbrack} & \; \\\left. {\left. {\left. {\left. {\left. {\left. {\left. {\left. \left. \left. {\left. \left| 0 \right. \right\rangle \otimes} \middle| \psi \right\rangle\mapsto \right. \middle| 0 \right\rangle \otimes \frac{U_{0m} + U_{1m}}{2}} \middle| \psi \right\rangle +} \middle| 1 \right\rangle \otimes \frac{U_{0m} - U_{1m}}{2}} \middle| \psi \right\rangle = \left. {\left. \quad\left| 0 \right. \right\rangle \otimes a_{m}} \middle| \psi \right.} \right\rangle +} \middle| 1 \right\rangle \otimes a_{m}^{\dagger}} \middle| \psi \right\rangle \equiv} \middle| \Phi_{m} \right\rangle & {{Eqn}.\mspace{14mu} 16}\end{matrix}$

That is, when the auxiliary output value of the auxiliary qubit AQ isobserved to be 0, the product (Equation 13) of the input value of thetarget qubit TQ (wave function shown in Equation 11) and the electronicannihilation operator a is output (first target output value). When theauxiliary output value of the auxiliary qubit AQ is observed to be 1,the product (Equation 12) of the input value of the target qubit TQ(Equation 11) and the electronic creation operator a is output (secondtarget output value). That is, regardless of whether the auxiliaryoutput value of the auxiliary qubit AQ is 0 or 1 here, a value usefulfor deriving the Green's function GF is output as the target outputvalue.

Furthermore, by applying quantum phase estimation, which is an algorithmfor quantum calculation of the eigenvalues of the unitary matrix, to theobtained Equations 12 and 13, each part of the Green's function GF canbe derived as represented by Equations 9 and 10. Further, by applyingEquation 17 to the Green's function GF, a quasiparticle spectrum can beobtained as shown in FIG. 7. FIG. 7 is a graph showing the calculationresult of the quasiparticle spectrum of the H_2O molecule.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 17} \right\rbrack & \; \\{{A(\omega)} = {{- \frac{1}{\pi}}{{Im}\left\lbrack {{tr}\left( {G\left( {\omega + {i\delta}} \right)} \right)} \right\rbrack}}} & {{Eqn}.\mspace{14mu} 17}\end{matrix}$

To summarize the above, preferably, in this quantum calculation method,in the calculation step, when the auxiliary output value is observed as0 in the observation step, the input value of the target qubit TQ andthe first linear sum O1 of the unitary operators U The product of and isoutput as the first target output value. In the calculation step, whenthe auxiliary output value is observed as 1 in the observation step, theproduct of the input value of the target qubit TQ and the second linearsum O2 of the unitary operators U is output as the second target outputvalue.

More preferably, this quantum calculation method further comprises aquantum phase estimation step. In the quantum phase estimation step,quantum phase estimation is performed for the first and second targetoutput values. In the specification step, a predetermined function isspecified based on the result of quantum phase estimation.

More specifically, referring to the quantum gate 53 b (diagonalcomponent) in FIG. 6, the auxiliary qubit AQ0 is set to the input value0, and the initialization state in which 0 and 1 are superpositioned asthe basis through the Hadamard gate is realized. Next, unitarytransformation by the unitary operators U_{0m} is executed when theauxiliary qubit AQ0 is 0. Subsequently, when the auxiliary qubit AQ0 is1, unitary transformation by the unitary operators U_{1m} is executed.After that, the observation of the output value is executed for theauxiliary qubit AQ0 through the Hadamard gate again.

With reference to the quantum gate 53 c (off-diagonal component) in FIG.6, the auxiliary qubit AQ0 and the auxiliary qubit AQ1 are set to theinput values 0, respectively, and the initialization state in which 0and 1 are superpositioned as the basis through the Hadamard gate isrealized. Next, unitary transformation by the unitary operators U_{0m}is executed for the target qubit TQ when the auxiliary qubit AQ1 is 0and the auxiliary qubit AQ0 is 0. Subsequently, when the auxiliary qubitAQ1 is 0 and the auxiliary qubit AQ0 is 1, unitary transformation by theunitary operators U_{1m} is executed for the target qubit TQ.Subsequently, a rotation of the phase π/4 is executed for the auxiliaryqubit AQ1.

Subsequently, when the auxiliary qubit AQ1 is 1 and the auxiliary qubitAQ0 is 0, unitary transformation by the unitary operators U_{0m′} isexecuted for the target qubit TQ. Subsequently, when the auxiliary qubitAQ1 is 1 and the auxiliary qubit AQ0 is 1, unitary transformation by theunitary operators U_{1m′} is executed for the target qubit TQ.

After that, the observation of the output value is executed for theauxiliary qubit AQ1 and the auxiliary qubit AQ0 through the Hadamardgate again.

3.4 Application to the Derivation of the Linear Response Function LF

Next, the linear response function LF, which is frequently used inmaterial calculation, will be described as an example. FIG. 8 is acircuit diagram for deriving the electronic density operator n of thelinear response function LF. FIG. 9 is a circuit diagram for derivingthe spin density operator s of the linear response function LF.

The linear response function LF includes a spin spin-spin responsefunction LF1 (Equation 18), a charge-charge response function LF2(Equation 19), and a charge-spin response function LF3 (Equation 20).

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 18} \right\rbrack & \; \\{L_{{\lambda{pj}},{p^{\prime}j^{\prime}}} \equiv S_{{\lambda{pj}},{p^{\prime}j^{\prime}}} \equiv {\left\langle \Psi_{gs} \middle| s_{pj} \middle| \Psi_{\lambda} \right\rangle\left\langle \Psi_{\lambda} \middle| s_{p^{\prime}j^{\prime}} \middle| \Psi_{gs} \right\rangle}} & {{Eqn}.\mspace{14mu} 18} \\\left\lbrack {{Equation}\mspace{14mu} 19} \right\rbrack & \; \\{L_{{\lambda{pn}},{p^{\prime}n}} \equiv {\sum\limits_{\sigma,{\sigma^{\prime} = \alpha},\beta}{N_{{\lambda p\sigma},{p^{\prime}\sigma^{\prime}}}\left( {N_{{\lambda p\sigma},{p^{\prime}\sigma^{\prime}}} \equiv {\left\langle \Psi_{gs} \middle| n_{p\sigma} \middle| \Psi_{\lambda} \right\rangle\left\langle \Psi_{\lambda} \middle| n_{p^{\prime}\sigma^{\prime}} \middle| \Psi_{gs} \right\rangle}} \right)}}} & {{Eqn}.\mspace{14mu} 19} \\\left\lbrack {{Equation}\mspace{14mu} 20} \right\rbrack & \; \\{L_{{\lambda{pn}},{p^{\prime}n}} \equiv {\sum\limits_{{\sigma = \alpha},\beta}{M_{{\lambda p\sigma},{p^{\prime}\sigma}}\left( {M_{{\lambda pj},{p^{\prime}\sigma^{\prime}}} \equiv {\left\langle \Psi_{gs} \middle| s_{pj} \middle| \Psi_{\lambda} \right\rangle\left\langle \Psi_{\lambda} \middle| n_{p^{\prime}\sigma^{\prime}} \middle| \Psi_{gs} \right\rangle} \equiv M_{{{\lambda p}^{\prime}\sigma^{\prime}},{pj}}^{*}} \right)}}} & {{Eqn}.\mspace{14mu} 20}\end{matrix}$

Further, the electronic density operator n and the spin density operators are represented as in Equation 21.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 21} \right\rbrack & \; \\{\;\left\{ \begin{matrix}{n_{p\sigma} = {a_{p\sigma}^{\dagger}a_{p\sigma}}} \\{s_{px} = \frac{{a_{p\alpha}^{\dagger}a_{p\beta}} + {a_{p\beta}^{\dagger}a_{p\alpha}}}{2}} \\{s_{py} = \frac{{{- {ia}_{p\alpha}^{\dagger}}a_{p\beta}} + {{ia}_{p\beta}^{\dagger}a_{p\alpha}}}{2}} \\{s_{pz} = \frac{{a_{p\alpha}^{\dagger}a_{p\alpha}} - {a_{p\beta}^{\dagger}a_{p\beta}}}{2}}\end{matrix} \right.} & {{Eqn}.\mspace{14mu} 21}\end{matrix}$

Here, the charge-charge response function LF2 will be further describedas an example. Same as the Green's function GF described above, theelectronic creation operator a+ and the electronic annihilation operatora can be represented by the linear sum O of the unitary operators U asshown in Equation 22.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 22} \right\rbrack & \; \\\left\{ \begin{matrix}{a_{p\sigma}^{\dagger} = \frac{U_{0{p\sigma}} - U_{1{p\sigma}}}{2}} \\{a_{p\sigma} = \frac{U_{0{p\sigma}} + U_{1{p\sigma}}}{2}}\end{matrix} \right. & {{Eqn}.\mspace{14mu} 22}\end{matrix}$

Here the unitary operators U as shown in Equation 23 is newly defined.

[Equation 23]

U _(κκ′σ) ^((p)) =U _(κpσ) U _(κ′pσ) (κ,κ′=0,1)  Eqn. 23

By using the Equation 23, the electronic density operator n can beexpressed as the Equation 24.

$\begin{matrix}{\mspace{79mu}\left\lbrack {{Equation}\mspace{14mu} 24} \right\rbrack} & \; \\{n_{p\sigma} = {{a_{p\sigma}^{\dagger}a_{p\sigma}} = {{\frac{U_{0{p\sigma}} - U_{1{p\sigma}}}{2}\frac{U_{0{p\sigma}} + U_{1{p\sigma}}}{2}} = \frac{U_{00\sigma}^{(p)} + U_{01\sigma}^{(p)} - U_{10\sigma}^{(p)} - U_{11\sigma}^{(p)}}{4}}}} & {{Eqn}.\mspace{14mu} 24}\end{matrix}$

The quantum gate 53 for calculating the linear sum O of such a unitaryoperators U is shown in FIG. 8. Specifically, it is preferable to definethe quantum gate 53 d for the diagonal component and the quantum gate 53e for the off-diagonal component. In particular, the diagonal componentsare represented as in Equation 25.

$\begin{matrix}{\mspace{79mu}\left\lbrack {{Equation}\mspace{14mu} 25} \right\rbrack} & \; \\{{\left. {\left. {\left. {{\left. {\left. {{\left. {{{{\left. {{\left. {\left. {{{\left. {\left. {{{\left. {\left. \left. {\overset{Ancilla}{\overset{︷}{\left. {\left. {\left. {\left| q_{1}^{A} \right. = 0} \right\rangle \otimes} \middle| q_{0}^{A} \right. = 0} \right\rangle}} \otimes \overset{Register}{\overset{︷}{\left. \left| \psi \right. \right\rangle}}}\mapsto \right. \middle| 0 \right\rangle \otimes} \middle| 0 \right\rangle \otimes \underset{= \mspace{14mu} 0}{\underset{︸}{\frac{U_{00\sigma}^{(p)} + U_{01\sigma}^{(p)} - U_{10\sigma}^{(p)} - U_{11\sigma}^{(p)}}{4}}}}{\quad\quad}\left. \quad\left| \psi \right. \right\rangle} +} \middle| 0 \right\rangle \otimes} \middle| 1 \right\rangle \otimes \underset{= \mspace{14mu}{\overset{\sim}{n}}_{p\sigma}}{\underset{︸}{\frac{U_{00\sigma}^{(p)} + U_{01\sigma}^{(p)} - U_{10\sigma}^{(p)} - U_{11\sigma}^{(p)}}{4}}}}\left. \quad\left| \psi \right. \right\rangle} +} \middle| 1 \right\rangle \otimes} \middle| 0 \right\rangle \otimes \underset{\text{?}}{\underset{︸}{\frac{U_{00\sigma}^{(p)} + U_{01\sigma}^{(p)} - U_{10\sigma}^{(p)} - U_{11\sigma}^{(p)}}{4}}}}{\left. \quad\left. {\left. \quad\left| \psi \right. \right\rangle +} \middle| 1 \right. \right\rangle \otimes}} \middle| 1 \right\rangle \otimes}\quad}\underset{\text{?}}{\underset{︸}{\frac{U_{00\sigma}^{(p)} + U_{01\sigma}^{(p)} - U_{10\sigma}^{(p)} - U_{11\sigma}^{(p)}}{4}}}\left. \quad\left| \psi \right. \right\rangle} = \left| 0 \right.} \right\rangle \otimes \left. \quad\left| 1 \right. \right\rangle \otimes {\overset{\sim}{n}}_{p\sigma}}{\quad\quad}} \middle| \psi \right\rangle + {\quad\quad}} \middle| 1 \right\rangle \otimes}\quad} \middle| 0 \right\rangle \otimes n_{p\sigma}} \middle| \psi \right\rangle \equiv} \middle| \Phi_{p\sigma} \right\rangle = {1 - n_{p\sigma}}}{\text{?}\text{indicates text missing or illegible when filed}}} & {{Eqn}.\mspace{14mu} 25}\end{matrix}$

Further, the charge-charge response function LF2 (Equation 19) can bederived by applying the quantum phase estimation as same as in the caseof obtaining the Green's function GF. Further, by applying Equation 26,a light absorption spectrum can be obtained.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 26} \right\rbrack & \; \\{\;{{\sigma(\omega)} = {{- \frac{4\pi}{c}}{{{Im}{Tr}}_{\chi}(\omega)}}}} & {{Eqn}.\mspace{14mu} 26}\end{matrix}$

Of course, since the spin density operator s can also be represented bythe linear sum O of the unitary operators U, the spin-spin responsefunction LF1 can be derived by performing quantum computation as thesame as in the case of the electronic density operator n.

More specifically, referring to the quantum gate 53 d (diagonalcomponent) in FIG. 8, the auxiliary qubit AQ0 and the auxiliary qubitAQ1 are set to the input values 0, respectively, and the initializationstate in which 0 and 1 are superpositioned as the basis is realizedthrough the Hadamard gate. Next, when the auxiliary qubit AQ1 is 0 andthe auxiliary qubit AQ0 is 0, unitary transformation by the unitaryoperators U_{00σ} is executed for the target qubit TQ. Then, when theauxiliary qubit AQ1 is 0 and the auxiliary qubit AQ0 is 1, unitarytransformation by the unitary operators U_{01σ} is executed for thetarget qubit TQ. Then, when the auxiliary qubit AQ1 is 1 and theauxiliary qubit AQ0 is 0, unitary transformation by the unitaryoperators U_{10σ} is executed for the target qubit TQ. Subsequently,when the auxiliary qubit AQ1 is 1 and the auxiliary qubit AQ0 is 1,unitary transformation by the unitary operators U_{11σ} is executed forthe target qubit TQ.

After that, the observation of the output value is executed for theauxiliary qubit AQ1 and the auxiliary qubit AQ0 through the Hadamardgate again.

Referring to the quantum gate 53 e (off-diagonal component) of FIG. 8,the auxiliary qubit AQ0 to the auxiliary qubit AQ2 are set to the inputvalues 0, respectively, and the initialization state in which 0 and 1are superpositioned as the basis is realized through the Hadamard gate.Next, when the auxiliary qubit AQ2 is 0, the first unitarytransformation is executed for the auxiliary qubit AQ0, the auxiliaryqubit AQ1, and the target qubit TQ. Next, when the auxiliary qubit AQ2is 1, the second unitary transformation is executed for the auxiliaryqubit AQ0, the auxiliary qubit AQ1, and the target qubit TQ.Subsequently, a rotation of the phase π/4 is executed for the auxiliaryqubit AQ2.

After that, the observation of the output value is executed for theauxiliary qubits AQ0 to AQ2 through the Hadamard gate again.

The quantum gate 53 used when deriving the spin-spin response functionLF1 is referred to FIG. 9. Specifically, the quantum gate 53 f isdefined for the diagonal component, and the quantum gate 53 g is definedfor the off-diagonal component. Further, the charge-spin responsefunction LF3 can be derived by the electronic density operator n and thespin density operator s. Furthermore, by using such a linear responsefunction LF, the light absorption spectrum can be obtained as shown inFIG. 10. FIG. 10 is a graph showing the calculation result of the lightabsorption spectrum of the C_2 molecule.

According to such a quantum calculation method described in thissection, the linear sum of the unitary operators can be calculated withquantization, thus physical quantities and physical functions includingthe linear sum of unitary operators can be calculated instead of theclassical algorithm in such a quantum computer. Can be calculated. Thatis, it is possible to calculate the linear sum of the unitary operators,physical quantities and physical functions including the linear sum ofthe unitary operators faster than the conventional technique using theclassical algorithm.

3.5 Creation of Oracle

By using n qubits, it is possible to take 2{circumflex over ( )}n linearcombinations of different states. The state of n qubits can be specifiedonly by specifying 2{circumflex over ( )}n degrees of freedom. In otherwords, the state of n qubits can hold 2{circumflex over ( )}n pieces ofinformation at one time.

In n qubits, each qubit can be in a superposition state (taken at thesame time) of 0 and 1. The state of n qubits can be represented as asuperposition state of 2{circumflex over ( )}n states because each qubitcan take two states of 0 and 1. Further, the 2{circumflex over ( )}ncoefficients representing the weights of each state are generallycomplex numbers. The method of arbitrarily controlling 2{circumflex over( )}n coefficients is called oracle, which is an important elementaltechnique in quantum technology.

Here, the realization of oracle is represented by using the linear sum Oof the unitary operators U described above. As shown in Equation 27, thestate of n qubits can generally be represented as the linear combinationof 2{circumflex over ( )}n states. This coefficient is set as c.

[Equation 27]

|Ψ

=c _(0 . . . 00)|0 . . . 00

+c _(0 . . . 01)|0 . . . 01

+ . . . +c _(1 . . . 11)|1 . . . 11

  Eqn. 27

Subsequently, Equation 27 is transformed into Equation 28 by using theoperators.

[Equation 28]

|Ψ

=(c _(0 . . . 00) Û _(0 . . . 00) +c _(0 . . . 01) Û _(0 . . . 01) + . .. +c _(1 . . . 11) Û _(0 . . . 11))|α

  Eqn. 28

Here, the operator {U_λ} in Equation 28 (λ is a number sequence in whichn 0s and 1s are arranged) is defined as Equation 29. In other words, itis defined as a unitary operator that acts on the state of a certainquantum qubit string a and converts it into the state of a quantum qubitstring called λ.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 29} \right\rbrack & \; \\\left. {\left. {\left. {\left. {\hat{U}}_{0{\ldots 00}} \middle| \alpha \right\rangle = \left| {0{\ldots 00}} \right.} \right\rangle\mspace{115mu}\vdots\left. {\hat{U}}_{1{\ldots 11}} \middle| \alpha \right.} \right\rangle = \left| {1{\ldots 11}} \right.} \right\rangle & {{Eqn}.\mspace{14mu} 29}\end{matrix}$

Here, Equation 28 means that the oracle can be written as a linearcombination of the unitary operator {U_λ}. That is, it means that anarbitrary oracle can be created by using the sum formula related to thelinear sum O of the unitary operators U described above.

As an example, considering that a state in which all qubits are 0 as acertain state a. That is, the oracle can be represented as Equation 30.Here, the operator {U_λ} can specifically configure a quantum circuit asshown in FIG. 11. FIG. 11 shows an example of various quantum circuitsrelated to oracle.

$\begin{matrix}{\mspace{79mu}\left\lbrack {{Equation}\mspace{14mu} 30} \right\rbrack} & \; \\\left. {\left. {\left. {\left. {{\left. \left| \Psi \right. \right\rangle = {\left( {{c_{0{\ldots 00}}{\hat{U}}_{0{\ldots 00}}} + {c_{0{\ldots 01}}{\hat{U}}_{0{\ldots 01}}} + \ldots + {c_{1{\ldots 11}}{\hat{U}}_{1{\ldots 11}}}} \right)\left. \quad\left| {0{\ldots 00}} \right. \right\rangle}}\mspace{79mu}\left. {\hat{U}}_{0{\ldots 01}} \middle| {0{\ldots 00}} \right.} \right\rangle = \left| {0{\ldots 01}} \right.} \right\rangle\mspace{220mu}\vdots\mspace{79mu}\left. {\hat{U}}_{1{\ldots 11}} \middle| {0{\ldots 00}} \right.} \right\rangle = \left| {1{\ldots 11}} \right.} \right\rangle & {{Eqn}.\mspace{14mu} 30}\end{matrix}$

That is, it is a quantum circuit in which X gate operation (Paulimatrices of rotation in the x-axis direction) is applied to the qubitsto be converted from 0 to 1 among the n qubits. The X gate operation isa gate operation that returns 1 when 0 is input and 0 when 1 is input.The operator {U_λ} can be specifically written down by combining X-gateoperations.

Given these concrete quantum circuit representations of the operator{U_λ}, we can actually create an arbitrary oracle by using the sumformula.

4. Others

In this section, a modified example of the quantum computer 1 will bedescribed. That is, the quantum computer 1 according to the presentembodiment may be further creatively devised according to the followingaspects.

(1) A program that causes the computer to function as the quantumcomputer 1 may be implemented.

(2) The quantum gate 53 may be implemented independently as a quantumcircuit.

Further, the present invention may be provided by each of the followingembodiments.

The quantum computer, wherein: the parameter group of k-th layerincludes 2 to the (n−k)-th power parameters.

The quantum computer, wherein: assuming that each coefficient in thelinear sum of the unitary operators is identified in order by using anindex represented by an n-digit binary number, then the setting unit isconfigured to set the parameters so that parameters included in theparameter group of the k-th layer depend on the ratio of a specific setof coefficients among each of the coefficients, wherein the specific setof coefficients are two coefficients having indices which differ fromeach other only in the k-th lowest digit, while the other digitscoincide with each other.

The quantum computer, wherein: assuming that each parameter included inthe parameter group of the k-th layer is identified in order by using anindex represented by an (n−k)-digit binary number, then the setting unitis configured to set the parameters so that a first parameter includedin the parameter group of the k-th layer depend on the ratio of cosinesfor a particular set of second parameters included in the parametergroup of the (k−1)th layer, wherein the particular set of the secondparameters consists of two parameters, each of which has an index with 0or 1 appended to the end of that of the first parameter.

The quantum computer, wherein: the quantum gate includes control unitarygates of n layers, the control unitary gate of k-th (2≤k≤n) layer inputsone control qubit, k−1 controlled qubits, and m target qubits, andincludes first and second control unitary gates of the (k−1)-th layerand a rotation gate, the first control unitary gate is configured tooperate the value of the controlled qubits when the value of the controlqubits is 0, the second control unitary gate is configured to operatethe value of the controlled qubits when the value of the control qubitsis 1, and the rotation gate applies on the control qubit.

The quantum computer, wherein: n=2.

The quantum computer, further comprising: an observation unit configuredto observe auxiliary output values, which are output values of the nauxiliary qubits at the quantum gate, wherein: the quantum gate isconfigured to output the product of an input value of the target qubitsand the linear sum of the unitary operators as a target output valuewhich is an output value of the target qubits when the auxiliary outputvalues observed by the observation unit are all 0.

The quantum computer, wherein: the quantum gate is configured to outputthe product of an input value of the target qubit and a first linear sumof the unitary operators as a first target output value when theobservation unit observes the auxiliary output values as 0, and outputthe product of the input value of the target qubit and a second linearsum of the unitary operators as a second target output value when theobservation unit observes the auxiliary output values as 1, wherein thesecond linear sum a Hermitian conjugate of the first linear sum.

The quantum computer, wherein: the first linear sum is an electronicannihilation operator in a predetermined function, the second linear sumis an electronic creation operator in the predetermined function, wherethe predetermined function is a Green's function or a linear responsefunction, further comprising a quantum phase estimation unit configuredto execute quantum phase estimation for the first and second targetoutput values, and wherein: the specification unit is configured tospecify the predetermined function based on a result of the quantumphase estimation.

The quantum computer, wherein: the quantum gate is configured to outputa state of any of the qubits by outputting the product of the inputvalue of the target qubits and the linear sum of the unitary operatorsas a standard output when the observation unit observes the auxiliaryoutput values as 0.

A non-transitory computer readable media storing a program, wherein: theprogram allows a computer to function as the quantum computer.

A quantum calculation method, comprising: a setting step to set aparameter group of n layers based on each coefficient in a linear sum ofunitary operators whose number is 2 to the n-th power, wherein theparameter group of k-th (2≤k≤n) layer is recursively set based on theparameter group of (k−1)-th layer; a calculation step to calculate apredetermined calculation on an input value input to each qubit based onparameters included in the parameter group of n layers, wherein each ofthe qubits has n+m qubits including n auxiliary qubits and m targetqubits; and a specification step to specify a linear sum of the unitaryoperators based on a calculation result in the calculation step.

The quantum calculation method, wherein: the parameter group of k-thlayer includes 2 to the (n−k)-th power parameters.

The quantum calculation method, wherein: assuming that each coefficientin the linear sum of the unitary operators is identified in order byusing an index represented by an n-digit binary number, then the settingstep sets the parameters so that parameters included in the parametergroup of the k-th layer depend on the ratio of a specific set ofcoefficients among each of the coefficients, wherein the specific set ofcoefficients are two coefficients having indices which differ from eachother only in the k-th lowest digit, while the other digits coincidewith each other.

The quantum calculation method, wherein: assuming that each parameterincluded in the parameter group of the k-th layer is identified in orderby using an index represented by an (n−k)-digit binary number, then thesetting step sets the parameters so that a first parameter included inthe parameter group of the k-th layer depend on the ratio of cosines fora particular set of second parameters included in the parameter group ofthe (k−1)th layer, wherein the particular set of the second parametersconsists of two parameters, each of which has an index with 0 or 1appended to the end of that of the first parameter.

The quantum calculation method, wherein: the calculation unit includescontrol unitary gates of n layers, the control unitary gate of k-th(2≤k≤n) layer inputs one control qubit and k−1 controlled qubits, andincludes first and second control unitary gates of the (k−1)-th layerand a rotation gate, the first control unitary gate is configured tooperate the value of the controlled qubits when the value of the controlqubits is 0, the second control unitary gate is configured to operatethe value of the controlled qubits when the value of the control qubitsis 1, and the rotation gate applies on the control qubit.

The quantum calculation method, wherein: n=2.

The quantum calculation method, further comprising: an observation stepto auxiliary outputs values that are output values of the n auxiliaryqubits at the quantum gate, wherein: the observation step outputs theproduct of an input value of the target qubit and the linear sum of theunitary operators as a target output value which is an output value ofthe target qubit when the auxiliary output values observed by theobservation step are all 0.

The quantum calculation method, wherein: the calculation step outputsthe product of an input value of the target qubit and a first linear sumof the unitary operators as a first target output value when theobservation step observes the auxiliary output values as 0, and outputsthe product of the input value of the target qubit and a second linearsum of the unitary operators as a second target output value when theobservation step observes the auxiliary output values as 1, wherein: thesecond linear sum a Hermitian conjugate of the first linear sum.

The quantum calculation method, wherein: the first linear sum is anelectronic annihilation operator in a predetermined function, the secondlinear sum is an electronic creation operator in the predeterminedfunction, wherein the predetermined function is a Green's function or alinear response function, a quantum phase estimation step is furthercomprised to execute quantum phase estimation for the first and secondtarget output values is further comprised, and the specification stepspecifies the predetermined function based on a result of the quantumphase estimation.

The quantum calculation method, wherein: the calculation step outputs astate of any of the qubits by outputting the product of the input valueof the target qubit and the linear sum of the unitary operators as astandard output when the observation step observes the auxiliary outputvalues as 0.

A quantum circuit, comprising: an input unit having n+m qubits includingn auxiliary qubits and m target qubits; a calculation unit configured tocalculate a predetermined calculation on an input value input from theinput unit based on the parameter group of n layers, wherein theparameter group of n layers is determined based on each coefficient inthe linear sum of unitary operators whose number is 2 to (n-th) power,and the parameter group of k-th (2≤k≤n) layer is recursively set basedon the parameter group of (k−1)-th layer; and a target qubits outputunit configured to output the product of the input value input to thetarget qubits and the linear sum of the unitary operators.

The quantum circuit, wherein: the parameter group of k-th layer includes2 to the (n−k)-th power parameters.

The quantum circuit, wherein: assuming that each coefficient in thelinear sum of the unitary operators is identified in order by using anindex represented by an n-digit binary number, then parameters includedin the parameter group of the k-th layer are determined so that each ofthem depends on the ratio of a specific set of coefficients among eachof the coefficients, wherein the specific set of coefficients are twocoefficients having indices which differ from each other only in thek-th lowest digit, while the other digits coincide with each other.

The quantum circuit, wherein: assuming that each parameter included inthe parameter group of the k-th layer is identified in order by using anindex represented by an (n−k)-digit binary number, then a firstparameter included in the parameter group of the k-th layer aredetermined so that each of them depends on the ratio of cosines for aparticular set of second parameters included in the parameter group ofthe (k−1)th layer, wherein the particular set of the second parametersconsists of two parameters, each of which has an index with 0 or 1appended to the end of that of the first parameter.

The quantum circuit, wherein: the calculation unit includes controlunitary gates of n layers, the control unitary gate of k-th (2≤k≤n)layer inputs one control qubit, k−1 controlled qubits, and m targetqubits, and includes first and second control unitary gates of the(k−1)-th layer and a rotation gate, the first control unitary gate isconfigured to operate the value of the controlled qubits when the valueof the control qubits is 0, the second control unitary gate isconfigured to operate the value of the controlled qubits when the valueof the control qubits is 1, and the rotation gate applies on the controlqubit.

The quantum circuit, wherein: n=2.

The quantum circuit, wherein: the target qubits output unit isconfigured to output the product when the observation results of theoutput values of the n auxiliary qubits are all 0.

The quantum circuit, wherein: the target qubits output unit isconfigured to output the product of an input value of the target qubitand a linear sum of a unitary operator so as to output the state of anarbitrary qubit when the output values of the n auxiliary qubits are allobserved to be 0.

Of course, the above embodiments are not limited thereto.

Finally, various embodiments of the present invention have beendescribed, but these are presented as examples and are not intended tolimit the scope of the invention. The novel embodiment can beimplemented in various other forms, and various omissions, replacements,and changes can be made without departing from the abstract of theinvention. The embodiment and its modifications are included in thescope and abstract of the invention and are included in the scope of theinvention described in the claims and the equivalent scope thereof.

1. A quantum computer, comprising: a setting unit configured to set aparameter group of n layers based on each coefficient in a linear sumof, unitary operators whose number is 2 to the n-th power, wherein theparameter group of k-th (2≤k≤n) layer is recursively set based on theparameter group of (k−1)-th layer; a quantum gate having n+m qubitsincluding n auxiliary qubits and m target qubits, and configured toexecute a predetermined calculation to each qubit based on the parametergroup of n layers; and a specification unit configured to specify thelinear sum of the unitary operators based on a calculation result of thequantum gate.
 2. The quantum computer of claim 1, wherein: the parametergroup of k-th layer includes 2 to the (n−k)-th power parameters.
 3. Thequantum computer of claim 2, wherein: assuming that each coefficient inthe linear sum of the unitary operators is identified in order by usingan index represented by an n-digit binary number, then the setting unitis configured to set the parameters so that parameters included in theparameter group of the k-th layer depend on the ratio of a specific setof coefficients among each of the coefficients, wherein the specific setof coefficients are two coefficients having indices which differ fromeach other only in the k-th lowest digit, while the other digitscoincide with each other.
 4. The quantum computer of claim 3, wherein:assuming that each parameter included in the parameter group of the k-thlayer is identified in order by using an index represented by an(n−k)-digit binary number, then the setting unit is configured to setthe parameters so that a first parameter included in the parameter groupof the k-th layer depend on the ratio of cosines for a particular set ofsecond parameters included in the parameter group of the (k−1)-th layer,wherein the particular set of the second parameters consists of twoparameters, each of which has an index with 0 or 1 appended to the endof that of the first parameter.
 5. The quantum computer of claim 1,wherein: the quantum gate includes control unitary gates of n layers,the control unitary gate of k-th (2≤k≤n) layer inputs one control qubit,k−1 controlled qubits, and m target qubits, and includes first andsecond control unitary gates of (k−1)-th layer and a rotation gate, thefirst control unitary gate is configured to operate the value of thecontrolled qubits when the value of the control qubits 0, the secondcontrol unitary gate is configured to operate the value of thecontrolled qubits when the value of the control qubits 1, and therotation gate applies on the control qubit.
 6. The quantum computer ofclaim 5, wherein: n=2.
 7. The quantum computer of claim 1, furthercomprising: an observation unit configured to observe auxiliary outputvalues, which are output values of the n auxiliary qubits at the quantumgate, wherein: the quantum gate is configured to output the product ofan input value of the target qubits and the linear sum of the unitaryoperators as a target output value which is an output value of thetarget qubits when the auxiliary output values observed by theobservation unit are all
 0. 8. The quantum computer of claim 7, wherein:the quantum gate is configured to output the product of an input valueof the target qubit and a first linear sum of the unitary operators as afirst target output value when the observation unit observes theauxiliary output values as 0, and output the product of the input valueof the target qubits qubit and a second linear sum of the unitaryoperators as a second target output value when the observation unitobserves the auxiliary output values as 1, wherein the second linear suma Hermitian conjugate of the first linear sum.
 9. The quantum computerof claim 8, wherein: the first linear sum is an electronic annihilationoperator in a predetermined function, the second linear sum is anelectronic creation operator in the predetermined function, where thepredetermined function is a Green's function or a linear responsefunction, further comprising a quantum phase estimation unit configuredto execute quantum phase estimation for the first and second targetoutput values, and wherein: the specification unit is configured tospecify the predetermined function based on a result of the quantumphase estimation.
 10. The quantum computer of claim 7, wherein: thequantum gate is configured to output a state of any of the qubits byoutputting the product of the input value of the target qubits and thelinear sum of the unitary operators as a standard output when theobservation unit observes the auxiliary output values as
 0. 11. Anon-transitory computer readable media storing a program, wherein: theprogram allows a computer to function as the quantum computer ofclaim
 1. 12. A quantum calculation method, comprising: a setting step toset a parameter group of n layers based on each coefficient in a linearsum of unitary operators whose number is 2 to the n-th power wherein theparameter group of k-th (2≤k≤n) layer is recursively set based on theparameter group of (k−1)-th layer; a calculation step to calculate apredetermined calculation on an input value input to each qubit based onthe parameter group of n layers, wherein each of the qubits has n+mqubits including n auxiliary qubits and m target qubits; and aspecification step to specify a linear sum of the unitary operatorsbased on a calculation result in the calculation step.
 13. A quantumcircuit, comprising: an input unit having n+m qubits including nauxiliary qubits and m target qubits; a calculation unit configured tocalculate a predetermined calculation on an input value input from theinput unit based on the parameter group of n layers, wherein theparameter group of n layers is determined based on each coefficient inthe linear sum of unitary operators whose number is 2 to the n-th power,and the parameter group of k-th (2≤k≤n) layer is recursively set basedon the parameter group of (k−1)-th layer; and a target qubits outputunit configured to output the product of the input value input to thetarget qubits and the linear sum of the unitary operators.
 14. A quantumcircuit of claim 13, wherein: the parameter group of k-th layer includes2 to the (n−k)-th power parameters
 15. The quantum circuit of claim 14,wherein: assuming that each coefficient in the linear sum of the unitaryoperators is identified in order by using an index represented by ann-digit binary number, then parameters included in the parameter groupof the k-th layer are determined so that each of them depends on theratio of a specific set of coefficients among each of the coefficients,wherein the specific set of coefficients are two coefficients havingindices which differ from each other only in the k-th lowest digit,while the other digits coincide with each other.
 16. The quantum circuitof claim 15, wherein: assuming that each parameter included in theparameter group of the k-th layer is identified in order by using anindex represented by an (n−k)-digit binary number, then a firstparameter included in the parameter group of the k-th layer aredetermined so that each of them depends on the ratio of cosines for aparticular set of second parameters included in the parameter group ofthe (k−1)th layer, wherein the particular set of the second parametersconsists of two parameters, each of which has an index with 0 or 1appended to the end of that of the first parameter.
 17. The quantumcircuit of claim 13, wherein: the calculation unit includes controlunitary gates of n layers, the control unitary gate of k-th (2≤k≤n)layer inputs one control qubit, k−1 controlled qubits, and m targetqubits, and includes first and second control unitary gates of the(k−1)-th layer and a rotation gate, the first control unitary gate isconfigured to operate the value of the controlled qubits when the valueof the control qubits is 0, the second control unitary gate isconfigured to operate the value of the controlled qubits when the valueof the control qubits is 1, and the rotation gate applies on the controlqubit.
 18. The quantum circuit of claim 17, wherein: n=2.
 19. Thequantum circuit of claim 13, wherein: the target qubits output unit isconfigured to output the product when the observation results of theoutput values of the n auxiliary qubits are all
 0. 20. The quantumcircuit of claim 13, wherein: the target qubits output unit isconfigured to output the product of an input value of the target qubitsand a linear sum of a unitary operator so as to output the state of anarbitrary qubit when the output values of the n auxiliary qubits are allobserved to be 0.